Step 1

Given:

$f(x,y)=\{\begin{array}{ll}cxy& 0<x<2,0<y<x\\ 0& \text{other}\end{array}$

a. value of constant c,

${\int}_{0}^{2}{\int}_{0}^{x}cxydydx=1$

${\int}_{0}^{2}cx{\left[\frac{{y}^{2}}{2}\right]}_{0}^{x}dx=1$

$\frac{c}{2}{\int}_{0}^{2}x\times {x}^{2}dx=1$

$\frac{c}{2}{\left[\frac{{x}^{4}}{4}\right]}_{0}^{2}=1$

2c=1

$c=\frac{1}{2}$

Step 2

b.

we have given a joint pdf,

$f(x,y)=\{\begin{array}{ll}cxy& 0<x<2,0<y<x\\ 0& \text{other}\end{array}$

marginal pdf of x,

${f}_{1}\left(x\right)={\int}_{0}^{x}f(x,y)dy$

${f}_{1}\left(x\right)={\int}_{0}^{x}\frac{1}{2}xydy$

$f}_{1}\left(x\right)=\frac{1}{2}x\times {\left(\frac{{y}^{2}}{2}\right)}_{0}^{x$

${f}_{1}(x)=\{\begin{array}{ll}\frac{1}{4}{x}^{3}& 0<x<2\\ 0& \text{otherwise}\end{array}$

$E\left(x\right)={\int}_{0}^{2}x\times {f}_{1}\left(x\right)dx$

$E\left(x\right)={\int}_{0}^{2}\times \frac{{x}^{3}}{4}dx$

$E\left(x\right)=\frac{1}{4}{\left(\frac{{x}^{5}}{5}\right)}_{0}^{2}$

$E\left(x\right)=\frac{8}{5}$

Step 3

$E\left({x}^{2}\right)={\int}_{0}^{2}{x}^{2}\times {f}_{1}\left(x\right)dx$

$E\left({x}^{2}\right)={\int}_{0}^{2}{x}^{2}\times \frac{{x}^{}}{}$

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